Distance at redshift z=0.666

[PAllen]

He means that what is posted in other forums should stay in other forums. If there's a problem with it, there's no point in discussing it here since the person who posted it in the other forum is not here to discuss it with.

If you are confident that the calculation you posted in post #26 is correct, then you should post it as coming from you, not some random person on some other forum, and you should be prepared to defend it on your own.

But don't you want to give credit to an idea/analysis you are using? I certainly would want to give such credit.

 

[PeterDonis]

don't you want to give credit to an idea/analysis you are using?

If it's a valid reference, sure. But some random person's post on some other forum is not a valid reference.

 

[Jaime Rudas]

If you are confident that the calculation you posted in post #26 is correct, then you should post it as coming from you,

$1+z_1=\dfrac{a_1}{a_0}=a_1=1.666$

The radial component of the FLRW metric for a flat universe is:

$ds^2=-c^2 dt^2+a^2 dr^2$

For light $ds=0$ from which we obtain:

$dr=\dfrac{c \, dt}a$

$H=\dfrac{\dot a} a$

$\dfrac{da}{dt}=H \, a$

$dr=\dfrac{c \, da}{a^2 H}$

From the Friedman equation for a flat universe and $\Omega_R=0$ we obtain:

$H=H_0 \sqrt{\dfrac {\Omega_{M_0}}{a^3} + (1-\Omega_{M_0)}}$

$\displaystyle r=\dfrac c{H_0} \ \int \dfrac{da}{\sqrt{\Omega_{M_0} a + (1-\Omega_{M_0}) a^4}}$

For the case at hand:

$\displaystyle d_a=\dfrac c{H_0} \ \int_1^{a_r} \dfrac{da}{\sqrt{\Omega_{M_0} a + (1-\Omega_{M_0}) a^4}}=6.28$ Gly

The above would mean that if we were to now emit a signal towards a galaxy that is now at 6.28 Gly, that signal would be received in that galaxy at a redshift of z=0.666.

 

[Jaime Rudas]

For a redshift of z=0.66, I get a current proper distance along a line of constant cosmological time of about 6.3 GLy, [...]. Coincidence that 6.3 current distance is nearly the same as the time it took to get here.

For an expanding universe, it is impossible for the current proper distance to coincide with the time taken for the signal to arrive here. Precisely because of the expansion, the current proper distance is necessarily greater than the path traveled by the light.

 

[Halc]

Precisely because of the expansion, the current proper distance is necessarily greater than the path traveled by the light.

Exactly so. So my labeling it as coincidence was more "not enough time for the two values to differ much".

My chart is also misleading since it seems to be older than the discovery of dark energy. Look at the v=c/2 worldline in my post4. It curves left the whole way, but after about half the age of the universe, dark energy became dominant and that worldline should start curving right, putting the current proper distance to it far larger than what this old picture shows.

 

[Jaime Rudas]

Exactly so. So my labeling it as coincidence was more "not enough time for the two values to differ much".

Yes, but 6.3 Gyr isn't something like a "short time", even on cosmological scales.

My chart is also misleading since it seems to be older than the discovery of dark energy. Look at the v=c/2 worldline in my post4. It curves left the whole way, but after about half the age of the universe, dark energy became dominant and that worldline should start curving right, putting the current proper distance to it far larger than what this old picture shows.

No, it isn't. The curve marked as v=c/2 isn't a world line, but a curve that denotes the behavior of the Hubble parameter H, which is monotonically decreasing. The world lines are shown on the left side of the graph, where in the world line marked as $r_0=8$ Gly a slight change in curvature can be noticed about 6 Gly ago.

 

[Halc]

Yes, but 6.3 Gyr isn't something like a "short time", even on cosmological scales.


No, it isn't. The curve marked as v=c/2 isn't a world line, but a curve that denotes the behavior of the Hubble parameter H, which is monotonically decreasing.

I'm reading it all wrong then. OK, the worldlines of comoving objects are the blue dashed ones on the left.

That leaves the dash-dotted black one to the left labeled simply 'horizon'. It seems to be the particle horizon, the 'size of the visible universe' at any given time, meaning that line intersects the lower "today's horizon" worldline at t=today.

I was hoping they'd put the event horizon on that chart somewhere, but it's not there.
It should start out lower than the 'today's horizon' line, but curved more, crossing the particle horizon line 10 Gyr ago and ending up today about where that r=16Gyr worldline is.

Would also be nice if they labeled the scalefactor on the vertical axis rather than just putting years there twice.

 

[Jaime Rudas]

I'm reading it all wrong then. OK, the worldlines of comoving objects are the blue dashed ones on the left.

Yes, that's why I find Tamara Davis's graph in her doctoral thesis much clearer:

That leaves the dash-dotted black one to the left labeled simply 'horizon'. It seems to be the particle horizon, the 'size of the visible universe' at any given time, meaning that line intersects the lower "today's horizon" worldline at t=today.

Yes, that's exactly what you can easily see in the Davis graph if you compare the particle horizon curve with an imagined dotted black line corresponding to z~1100.

I was hoping they'd put the event horizon on that chart somewhere, but it's not there.
It should start out lower than the 'today's horizon' line, but curved more, crossing the particle horizon line 10 Gyr ago and ending up today about where that r=16Gyr worldline is.

Would also be nice if they labeled the scalefactor on the vertical axis rather than just putting years there twice.

All that and much more you can see very clearly in the Davis graph.